Table 2.

Density estimation for X₁ and X₂ from B=250 simulated data sets. Reliability ratios for X₁ and X₂ are 0.50 and 0.75, respectively. Statistics reported: (a) $MSE\left(W\right)∕MSE\left(\widehat{X}\right)$, where $MSE\left(\widehat{X}\right)=B^{-1}{\sum }_{b=1}^B{n}^{-1}{\sum }_{i=1}^n{({\widehat{X}}_{i,b}-X_{i,b})}^2$ (coefficient of variation ≈ 0.001), and (b) $ISE\left(G_W\right)∕ISE\left(G_{\widehat{X}}\right)$, where $ISE\left(G_{\widehat{X}}\right)=B^{-1}{\sum }_{b=1}^B\int {\left\{G_{\widehat{X},b}\right(t)-G_{X,b}(t\left)\right\}}^{2}dt$, for $G_X\left(t\right)=n^{-1}{\sum }_{i=1}^n{I}_{X_i\le t}$, −∞ < t < ∞ (coefficient of variation ≈ 0.02). Adjusted data $\widehat{X}$: RC, regression calibration; MR, moment reconstruction; M2 and M4, multivariate MAI with M=2 and 4, respectively; U2 and U4, univariate MAI with M=2 and 4, respectively.

					$\left(a\right)\frac{MSE\left(W\right)}{MSE\left(\widehat{X}\right)}$
Variable	ρ	n	RC	MR	M2	M4	U2	U4
X 1	0.5	1000	2.04	1.79	1.74	1.94	1.71	1.85
		2000	2.04	1.79	1.74	1.89	1.71	1.82
	0.7	1000	2.43	2.01	2.04	2.24	1.71	1.88
		2000	2.45	2.01	2.04	2.21	1.71	1.86
X 2	0.5	1000	1.33	1.26	1.23	1.36	1.22	1.35
		2000	1.33	1.27	1.23	1.36	1.22	1.35
	0.7	1000	1.30	1.20	1.23	1.37	1.22	1.35
		2000	1.30	1.23	1.23	1.37	1.22	1.35
					$\left(b\right)\frac{ISE\left(W\right)}{ISE\left(\widehat{X}\right)}$
Variable	ρ	n	RC	MR	M2	M4	U2	U4
X 1	0.5	1000	1.86	5.49	4.35	10.51	4.34	10.48
		2000	1.88	5.85	4.66	18.25	4.65	18.13
	0.7	1000	2.65	6.55	5.30	10.57	4.34	10.40
		2000	2.79	7.07	5.67	16.97	4.61	16.61
X 2	0.5	1000	1.36	2.73	2.27	6.48	2.27	6.43
		2000	1.44	2.93	2.42	8.48	2.43	8.36
	0.7	1000	1.37	2.78	2.29	7.10	2.32	7.05
		2000	1.37	2.98	2.38	8.73	2.41	8.62